
with \(e=g_1 g_2/\sqrt{g_1^2+g_2^2} = g_1 c_W = g_2 s_W\)  being the electric charge.
Moreover, taking into account the \(Z-Z^\prime \)  mixing in Eq. (REF ) and the vertex corrections [1]}, [2]}, we have the following modified \(Z\)  couplings to leptons
\(\begin{aligned}\Delta _{ij}^{\ell L} &= g_{\rm SM}^{\ell L}\left( \delta _{ij} + \sin \xi \frac{g_{ij}^{\ell L}}{g_{\rm SM}^{\ell L}} + \sum \limits _k \frac{g_{ik}^{\ell L} g_{kj}^{\ell L}}{(4\pi )^2} {K_F}\left( \frac{m_Z^2}{m_{Z^{\prime }}^2} \right) \right)\,,\\\Delta _{ij}^{\nu L} &= g_{\rm SM}^{\nu L}\left( {{\delta _{ij}} + \sin \xi \frac{{g_{ij}^{\ell L}}}{{g_{{\rm {SM}}}^{\nu L}}} + \sum \limits _k {\frac{{g_{ik}^{\ell L}g_{kj}^{\ell L}}}{{{{(4\pi )}^2}}}} {K_F}\left( {\frac{{m_Z^2}}{{m_{Z^{\prime }}^2}}} \right)} \right)\,,\\\Delta _{ij}^{\ell R} &= g_{\rm SM}^{\ell R}\left( {{\delta _{ij}} + \sin \xi \frac{{g_{ij}^{\ell R}}}{{g_{{\rm {SM}}}^{\ell R}}} + \sum \limits _k {\frac{{g_{ik}^{\ell R} g_{kj}^{\ell R}}}{{{{(4\pi )}^2}}}} {K_F}\left( {\frac{{m_Z^2}}{{m_{Z^{\prime }}^2}}} \right)} \right)\,,\end{aligned}\) 
